1. If 𝑔 is a twice differentiable function and 𝑓(𝑥) = 𝑒^𝑥^2 ∙ 𝑔(𝑥^2) , find 𝑓′ in terms of 𝑥, 𝑔, 𝑔′ , and 𝑔′′ .

2. Let 𝑟(𝑥) = 1/ (3𝑓(𝑔(𝑥) )^2 , where 𝑔(1) = 3 , 𝑔′ (1) = 9 , 𝑓(3) = 2, and 𝑓′ (3) = 6 , find 𝑟′ (1).

Community Answer

(1) Given,f(x)=e^(x^(2))*g(x^(2))Differentiating w.r. & x we get{:[f^(')(x)=e^(x^(2))*g^(')(x^(2))*2x+e^(x^(2))*2x*g(x^(2))],[[[(d)/(dx)(u*v)],[=u*(dv)/(dx)+v*(du)/(dx)]]],[=2xe^(x^(2))(g^(')(x^(2))+g(x^(2)))]:}Hence, the resultf^(')=2xe^(x^(2))(g^(')+g)(2) Given,r(x)=(1)/((3f(g(x))^(2))=(1)/(9)*(1)/((f(g(x)))^(2))Differentiating w.r. & x, we get{:[r^(')(x)=(1)/(g)((f(g(x))) ... See the full answer